Superimposing these will give a periodic function.
How can additional symmetries be achieved?

We define several useful isometries that are important
here: h is a horizontal half-translation, d a diagonal
half-translation, v a vertical half-translation
and R is a half-turn about the origin.

The general lattice cell

Each of these can be used as a negating symmetry to create a function
invariant under p1 but with additional anti-symmetries. Some turn
out to be algebraically equivalent. For instance, forcing h to be
negating is algebraically the same as making d negating , so these are not traditionally
thought of as being different.

Furthermore, we would not want to create functions with h, v,
or d as actual symmetries, because these would then be symmetric in
a finer lattice (one with a smaller cell). If the goal were to reduce the
lattice, why not just start with a smaller one to begin with?

Thus there is only one new pattern type that can be formed from the group
p1. For similar reasons, there is only one type that grows out of
p2, the one where R is negative.